Some General Properties of the Covariation Matrix for MIMO Communications Channels

Abstract

It is proposed the use of the prolate spheroidal wave functions (PSWF) as a universal basis in order to represent the covariation matrix function for the Multiple Input – Multiple Output (MIMO) communication channel.

Full text

Some General Properties of the Covariation Matrix for MIMO
Communication Channels
Alberto Alcocer–Ochoa, Valeri Ya. Kontorovitch,
Communications Section, Electrical Engineering Department,
CINVESTAV IPN, Mexico.
I. P. N. Ave 2508, San Pedro Zacatenco, 07360,
Mexico City, Mexico.
Phone: (+5255) 5061–3800, Ext.: 6367,
Fax: (+5255) 5061–3977.
e–mail: aalcocer@ieee.org, valeri@cinvestav.mx
Ramón Parra–Michel,
Electrical Engineering Department,
ITESM campus Guadalajara.
Gral. Ramón Corona Ave 2514, 45140,
Zapopan, Jalisco, Mexico.
Phone: (+5233) 3669–3000, Ext.: 3140,
Fax: (+5233) 3669–3061.
e–mail: rparra@itesm.mx
Abstract – It is proposed the use of the prolate spheroidal
wave functions (PSWF) as a universal basis in order to
represent the covariation matrix function for the Multiple
Input – Multiple Output (MIMO) communication channel.
Keywords – Channel covariation matrix, universal basis,
eigenvalues and eigenfunctions.
I. Introduction.
The emerging interest for MIMO channel modeling
and simulating has caused a strong attention to the
problems of channel covariance matrix calculations. There
are numerous papers already published where those
problems are considered [1–5]. But to the best of our
knowledge, the methodology, the calculus and the general
properties of the covariation matrix for MIMO channel are
still not well stated (see e. g. [6]) and are directly related to
the interpretation of the problems of scatters reflection and
refraction phenomenology of clusters of scatters and its
influence to the properties of the received signal.
The way to represent the system characteristics of the
channel by means of a finite set of artificial trajectories
based on the use of the Karhunen–Loève integral Equation
(KLE), was successfully applied for the scalar Single Input
– Single Output (SISO) case (see [7–8] and references
therein). The attempt to apply the same approach was
presented by the authors at [9] for MIMO channels.
It was shown at [7] that for the scalar case (SISO) the
effective way to avoid the solution of the KLE is to apply
auniversal basis1 for the representation of the system
characteristics of the channel, never mind if their
covariation functions are separable or not by its arguments
“t” and “IJ”. At [10–11] were proposed some basis for the
representation of the MIMO channel matrix H in order to
simplify the model for a large number of inputs and
outputs.
Hereafter we will tackle the problem on how to
calculate the correlation matrix for H and what can be a
universal basis for the MIMO case. It can also be
considered in the same sense as in [10–11]. A matrix–
valued wavelet expansion [12] can be used as well.
1 The idea behind a universal basis is to get a proper orthogonal
representation of the channel impulse response, which is invariant to the
channel covariation matrix (contrary to what happens with KLE). This is
why we said that this basis is universal, but for sure it is not unique.
It is worth to mention here that practically the same
problems for the covariation matrix calculation exist at the
radar theory (see e. g. [13–14]), where some basic
properties for the covariance matrix were proposed, and is
feasible to apply them for the MIMO channel covariance
matrix calculation. The purpose of this paper is also to
evoke those methods. Hereafter we do not consider
polarization effects (only the horizontal plane, i. e., a bi–
dimensional model), and all the assumptions made here
are the traditional for the macro– and microcell wireless
systems cases [1–2].
BS MS
BS
1
M
BS
2
M
BS
1
T
BS
2
TMS
2
T
MS
1
T
MS
1
M
MS
2
M
D
BS MS
BS
1
M
BS
2
M
BS
1
T
BS
2
TMS
2
T
MS
1
T
MS
1
M
MS
2
M
D
Fig. 1. The geometrically–based MIMO channel model.
We take into account the Geometrically–Based Wide–
Sense Stationary Uncorrelated Scattering (GBWSSUS)
MIMO channel model (see e. g., [15–17] and references
therein), which is depicted for the general case at Fig. 1. In
the model it is distinguished two kinds of scatters: the
local scatters, which are confined into a circular region at
the surrounding of the MS (BS); and the dominant
scatters, which are in the circular ring regions2 (as depicted
in Fig. 1). The matrix impulse response for this channel is
H(t, IJ,ș) with scalar components [2, 18–19]:
,)(),t(g
ˆ
),,t(h
ˆ
n,m
K
1k
n,m
k
n,m
kn,m ¦
WWGT TW (1)
where ƣk
m,n(t, ș) are the complex amplitudes received from
the k–th cluster and Km,n is the number of clusters.
Here we consider concretely one special case taken
form Fig. 1, which is depicted at Fig. 2. Each summand in
(1) can be represented in the following way, if at the
receiver point one use the l–th element of the linear
antenna array3 (see Fig. 2) (here the į(·) components are
skipped):
2 For generality, we consider both types of scatters at both system link
ends.
3 The generalization for other types of antennas can be done as well.

,ee)t(g
ˆ
)(G)(G
4
P
),t(h
ˆ
tj
sin)/x(j2
n,m
k
0kR0kT
k
0
k
n,m
n,m
k
kQ
TOS
D

TTMM
S
l
lr
K
(2)
where P0 is the average power at the transmitter, 1 Į 3
is the propagation exponent, ij0 and ș0 are the steering
Angle of Departure (AoD) and Angle of Arrival (AoA) for
the transmitter and receiver antennas, respectively. GT(·)
and GR(·) are the antenna patterns for the transmitter and
receiver, respectively. ƣk
m,n(t) is the received complex
amplitude, rk is the distance vector for the k–th cluster, xl
is the position of the l–th antenna element referred to the
geometrical center of the linear antenna array, Șl is its
radii–vector, Ȟk
m,n is the Doppler shift provided by the MS
movement. Note that if |ijk – ij0| are rather small for every
k, then GT(0) §const, and (2) can be simplified in the
following way:
,ee)t(g
ˆ
)(GPconst),t(h
ˆtj
sin)/x(j2
n,m
kkRk
k
n,m
n,m
k
kQ
TOS
T l
l
K(3)
where, without any lost in generality, we have defined șk =
șk – ș0, and Pk = P0 / (4ʌ|rk|)Į.
BS MS
T
M
D
r
BS MS
T
M
D
r
Fig. 2. A single bounce geometrical scenario.
The covariance function bk
l,ȡ of Ʃk
m,n for two antenna
elements l and ȡ, is bk
l,ȡ = <(Ʃk
m,n) (Ʃk
m,n)*>, where l,ȡ = 1,
2, …, K, and K is the number of antenna elements.
II. Structure of the Covariation Matrix.
Let us consider two elements “l” and “ȡ” for a linear
antenna, then the matrix covariance element bk
l,ȡ will stand
only for space components4 as:
.)(sinde)(W)(Gconstb
2
1
2
1
sin)xx)(/j2(
2
R
k
,l ³
'
'
TOS
TU TTT Ul(4)
From (4) it follows that bk
l,ȡ are the Fourier coefficients
for |GR(ș)|2 Wș(ș), which is defined for șmin șș
max, and
Wș(ș) is the AoA Probability Density Function (PDF), and
¨ș = |șmax – șmin|.
The structure of the matrix B = [bk
l,ȡ] depends on the
selection of the basis of element representation5, but taking
into account that ¨ș is a limited and rather narrow angle,
the eigenvalues for this matrix has an isolated group of
large values [13]. Although B can be actually
approximated in a subspace which dimension is equal to
4 Here are usually taken the assumptions for factorization of space–time
covariation matrixes (see [1–2, 13]). The part of the component of bkl,ȡ
which depends on the time variation can be found at [14, 29].
5 The simplest way for its representation is by doing it in its proper basis,
but it is a complex problem [13], e. g., from [4] it can be seen that a
representation can be done in a Bessel functions basis, but a series
expansion in this basis converges very slowly (see [30]). Meanwhile
when |GR(ș)|2§const and Wș(ș) = 1/2ʌ, from (4) follows the well know
Jakes’ model.
the one of this isolated group. Hence the effective rank
(reff) of the matrix B can be evaluated from [14], where
this problem was tackled by applying functions with
double orthogonality, concretely the PSWF as the basis
(see [20]). At [13] it was shown that:
¬¼
,1)2/(cr 0eff S (5)
where ¬¼ is the integer part operator, and c0 is a parameter
which can be found from the antenna aperture’s
normalization: ¨zl,ȡ = 2 (xȡ – xl) / Da, then [13] c0 = (ʌ Da /
O) (ijj – ij0), and
¬
¼
,1)2/(D)(r a0jeff OMM (6)
where Da is the dimension of the linear antenna array.
When the argument of ¬¼ is a small value, then reff § 1,
and it happens for many practical cases. It is well known
(see [14]), that when ¨ș/Gș » 1, where Gș§ 1/Da, the
PSWF can be approximated by sinc() functions, and reff
represents the number of uncorrelated samples in the
antenna aperture space domain. With these considerations,
asymptotically form (4) one gets:
>@
,ecos)xx(csinconstb sin)xx)(/j2(
k
,l
TOS
UU
U
T l
l(7)
which naturally coincides with [14, 21]. It can be seen
from (4) that for ¨ș small, [bk
l,ȡ] can be represented only
by one eigenfunction with ș§¢ș². Here const means the
product of all other parameters multiplied by ¨ș.
Though in the general case, the matrix B in its own
basis can be represented in a subspace with the span reff as
in [13]:
,
H
VVB /| (8)
where [V1], [V2], …, [Vr] are the column matrix (K x reff)
of eigenvectors, / is a diagonal matrix of the isolated
group of eigenvalues with rank reff. The properties of the
PSWF depend only on the antenna parameters and
qualitatively, the representation of B in the PSWF basis in
the form given by (8) can be considered as a universal
basis. For scalar time–dependent channels only, these
universal bases were proposed in [7, 22].
When the sinc() function in (7) is approximately the
unity (i. e., (xȡ – xl)cos(ș)o 0), then:
,econstb sin)xx)(/j2(
k
,l
TOS
U
U
 l and (9)
),()(constB 0
H
0TT VV (10)
where V(ș0) is the steering linear array antenna vector
with components . Certainly (9) and (10) are well known
and can be generalized as it is done in [4] for any type of
antenna array, with the appropriate form of the steering
vector. It happens in the asymptotic case that for any
argument of the PSWF, which depends on the antenna
geometry, the sinc() function tendency behavior preserves
(as it was shown beforehand) and formula (10) will be
valid, but the components for the (K+1) steering vector at
(10) will be obviously different.
Beforehand we assume that the AoDs are relatively
narrow, what it is not always the real scenario. For
example, for the scenario depicted in Fig. 3, the AoD

which is ¨ș = ș1
BS + ș2
BS is rather large as illumination is
provided for both rings of scatters, GT(șk – ș0) have to be
considered and the integral in (4) can be simplified in the
way:
.)(sinde)(W)(GRe
d)(W)(Gconstb
sin)xx)(/j2(
2
R
2
T
k
,l
¿
¾
½
¯
®
MMM
TTT|
³
³
<
MOS
M
4
TU
Ul
(11)
As the first term is a constant, which does not depends
on ij, one can see that the main properties of [bk
l,ȡ]
discussed at section II, as still valid here as well.
BS MS
BS
1
T
BS
2
T
MS
1
M
MS
2
M
D
BS MS
BS
1
T
BS
2
T
MS
1
M
MS
2
M
D
Fig. 3. Local and dominant scatters reflection scenario.
Regarding the case of reff § 1, it is necessary to
dedicate a few words to the maximum eigenvalue Omax. It
is worth to mention that formally, B(Q)lG(Z) by means
of a Fourier transform and if /0 = [0, T] x [0, T] x [0, R] is
the space of consideration, then asymptotically [14] :
,)(de)( j2
³
f
f
S OZ
Z
ZGQQB Q and
).(max
max Z
Z
G O
(12)
here Q is a vector argument for B() and Z is a vector
argument for its Fourier transform. Other methods for the
estimation of Omax can be found at the theory of matrixes
(see e. g. [23]). (For the proof of (12) see Appendix A.)
The expression (12) shows how the maximum
eigenvalue of B() can be evaluated through the Power
Density Spectrum (PDS), useful when reff § 1, and that
asymptotically the covariation matrix B has a continuum
spectrum of eigenfunctions and eigenvalues, and not the
discrete ones.
III. More about the Structure of the
Covariation Matrix.
Let us consider the scenario depicted at Fig. 3,
supposing that the MS occupy the position of the BS and
viceversa. This assumption gives us an opportunity to
introduce the mobility of the MS together with a more
general case of the scattering scenario: the MS mobility
was considered at [24], assuming only the movement of
the geometrical center of the MS scattering ring.
Here we will consider a different model: we will
neglect the changes in rk and the instant changes in the
Doppler shift Qk, but we will take into account the changes
in the cluster scenario with the MS movement. The
attempt to take all above mentioned phenomena into
account is done in [25] and is partially investigated in [26]
as a Spatial Channel Model (SCM). Here we make some
further simplifications to the SCM to get an analytical
solution.
The movement of the MS will include not only
changes in the number of active clusters, but changes of
the types of clusters: disappearing dominant (local)
clusters and appearing new local (dominant) ones. It can
be modeled by assuming that the MS randomly encounter
both of them, being in different states and applying only
local scatters or only dominant ones with a fast change6 of
these states. In the first state the BS will see the angle ij1
BS
and in the second ij1
BS + ij2
BS (see Fig. 3)7.
Taking into account the MS movement, both of those
angles are stochastic processes. Such kind of scenarios can
be successfully modeled by the models with random
structure (see [27, chp. 6]). Supposing that both angles
have approximately Gaussian distributions in the linear
aperture, one can use the results of example [27, pp. 312].
The essence of the modeling of the above mentioned
scenario with high intensity of the changes of those two
states is to create a new stochastic process ijfBS(t) which
has two components: ij1
BS(t) and ij2
BS(t) with known one–
dimensional distributions Wij(ij1
BS) and Wij(ij2
BS)
(Gaussian in our case). Using the results of the example in
[27, chp. 6], it is easy to obtain the two–dimensional PDF
for ijfBS(t) as:
,e
2
H
2
H
!n2
1
)(W)(W),(W
n
0n
BS
n
BS
n
n
BSBSBSBS
BSBS
WD
f
f
f
f
f
f
M
f
M
ff
f
W
W
W
W
ff
W
¦¸
¸
¹
·
¨
¨
©
§
V
M
¸
¸
¹
·
¨
¨
©
§
V
M
MM MM
(13)
where Vf2 = ½ (V1
2 + V2
2), Df = 2Qp1, and p1 = p2 are the
final probabilities for the states one and two, where
W(ijfBS) is a zero mean, and ıf2 variance normal
distribution, and Hn(x) are the Hermite polynomials.
Now applying (13) for the calculus of the covariation
term <(Ʃk
m,n (t1,Kl)) (Ʃk
m,n (t2,Kȡ))*> for the antenna
elements l and ȡ (see sections I and II), one can obtain:
.dcose)(H)(G
d)(H)(W
!n2
e
bconstb
BS
BS
BS
BS
BSBS
sin)xx)(/j2(
BS
n
2
BS
R
BSBS
n
BS
0n
n
n
k
,l
k
,l
³
³
¦
f
fU
W
f
WWW
W
f
f
W
M
ff
MOS
ff
M
fff
M
f
WD
UU
MMMM
MMM
l
(14)
The last integral is equal to one for n = 0 and for n > 0
is zero, though,
.dcose)(Gbconstb
BS
BS BSBS
sin)xx)(/j2(
2
BS
R
k
,l
k
,l ³
f
fU
W
M
ff
MOS
fUU MMM l
… (15)
Formula (15) (see Appendix B for details) shows that
the elements bk
l,ȡ are completely separable. From the
structure for bk
l,ȡ in (15), one can also see that the material
of section II is totally valid for this case as well.
6 Here we apply the word fast change in the same context as it was
considered in [27], i. e., the intensities of changes of the states are
sufficiently higher than Q§ 1/Wc, being Wc the covariation time in each
state.
7 Please note that the etiquettes are already changed.

IV. Comparison of the Universal PSWF
Eigenbasis with other Proposals for
Channel Orthogonalizations.
Here we will compare the PSWF Eigenbasis with the
so–called virtual MIMO channel representation [10] and
other basis, already used by some authors for the
orthogonal representation (see e. g. [30])8.
a) Comparison with [10]. For the lack of space, we can
not make a detailed review of [10], but we would like to
mention that actually the virtual channel representation is
a partitioning procedure of the spatial propagation
environment, into MN virtual AoD–AoA artificial
trajectories. Those trajectories are predefined by a
uniform quantization of the angles.
But considering, e. g., equations (1), (21), etc. from
[10], one can see that the functions,
,
sin
Qsin
Q
e
)(f
Qj2
QST
TS
T
TS
(16)
which are the basis for HV(q, p), (for definitions see the
original material at [10]), has a module,
,
sin
Qsin
Q
1
)(fQST
TS
T (17)
and it is nothing else but a beam pattern for a uniform
weighted linear antenna. It was shown that this basis can
be represented by PSWF (see pp. 384–385 [14]).
b) At [30] it was proposed a Salz–Winters model with
the Bessel series expansion for bl,U which converges very
slowly. At [21] this model was modified for the case of 'T
small and shown that for this case, bl,U can be represented
by only one sinc() function, but this is exactly an
asymptotic case for PSWF [13, 28].
V. Conclusions.
The paper provides a unique methodology for the
calculation of the principal element of the covariation
matrix for the MIMO channel, and introduces the idea of a
universal basis, by means of the PSWF, which has double
orthogonality properties. The PSWF are a complete basis
that does not depend on the channel properties, but it does
on the antenna parameters, as stated above
The calculation of the distribution functions for Wij(ij)
and Wș(ș) at (4) and (11) follows from the previous work
of the authors at [9, 18].
With the symmetry of the channel for the transmitter
and receiver terminals, the formulas (4), (11) and (15) are
valid for the covariation matrix at the transmission end, as
well.
VI. Appendixes.
Appendix A.
Let us consider the KLE [28]:
8 To the best of our knowledge, there are a few attempts to apply the
orthogonalization principle for the MIMO channel.
,'dr'd'dt)',','t()',','tt(B),,t( ³³³
f
f
f
f
f
f
WWWW WO rrrr <<
… (A.1)
this integral equation has a solution for the scalar case as
[28],
.r,,t,e)r,,t( rt
)rt(j rt fZWZZf W W
ZWZZ W
<(A.2)
By substituting (A.2) into (A.1) one can get the first
equation in (12).
Appendix B.
Let us consider the product <(Ʃk
m,n) (Ʃk
m,n)*> in the
way:



>@
^`
,
ee)t(g
ˆ
)t(g
ˆ
)(G)(GRe
)4(
)t(cosP
),t(h),t(h
)t(sinx)t(sinx)/j2(
)tt(j
21
2
RT
2
k
2
10
*
2
k
n,m1
k
n,m
21
12
n,m
kMMOS
Q
U
U
U
MT

S
M
l
l
lr
KK
… (C.1)
then,
 
>@
,
ddcose),(W)(G)(GRe
)t(g
ˆ
)t(g
ˆ
)4(
eP
),t(h),t(h
)t(sinx)t(sinx)/j2(
R
2
T
2
*
1
2
k
2
j
0
*
2
k
n,m1
k
n,m
21
n,m
k
°
¿
°
¾
½
°
¯
°
®
MMMMMMT

S
³³
MM
W
MMOS
W
T
U
WQ
U
W
Ul
ll r
KK
… (C.2)
Here we apply W = t2 – t1 and assume that cos(ij) does
not change sufficiently during the small interval W, because
the changes of the states are fast comparing with Wc§ 1/Q;
and for Wof, all the components of the covariation
matrix tends to zero.
Though the exponential term for small W can be
represented in the way
>
@
MWOSMOS UU 2sinx)/j2(sinxx)/j2( ee l, the
module of the second term for Wo 0 tends to one, hence
approximately its influence can be neglected and one can
easily obtain (15); here ij = ijf and ijW = ijfW:
.dcose)(Gbconstb
BS
BS BSBS
sin)xx)(/j2(
2
BS
R
k
,l
k
,l ³
f
fU
W
M
ff
MOS
fUU MMM l
where WQ
UU
W
n,m
k
j
2
*
1
k
,l e)t(g
ˆ
)t(g
ˆ
bl and T
T )(Gconst T.
VII. References.
[1]. Hao Xu, Dmitry Chizhik, Howard Huang & Reinaldo Valenzuela.
“A Generalized Space–Time Multiple–Input Multiple–Output
(MIMO) Channel Model”. IEEE Transactions on Wireless
Communications. Vol. 3, No. 3, Pages: 966 – 975, May 2004.
[2]. Andreas F. Molisch. “A Generic Model for MIMO Wireless
Propagation Channels in Macro– and Microcells”. IEEE
Transactions on Signal Processing. Vol. 52, No. 1, Pages: 61 – 71,
January 2004.
[3]. Shuangquan Wang, Kaustubha Raghukumary, Ali Abdi, Jon
Wallace & Michael Jensen. “Indoor MIMO Channels: A Parametric
Correlation Model and Experimental Results”. The IEEE/Sarnoff
Symposium on Advances in Wired and Wireless Communications
2004. Pages: 1 – 5, 26 – 27, April 2004.

[4]. Antonio Forenza & Robert W. Hearth Jr. “Impact of Antenna
Geometry on MIMO Communication in Indoor Clustered
Channels”. IEEE Symposium on Antennas and Propagation Society,
2004. Vol. 2, Pages: 1700 – 1703, 20 – 25 June 2004.
[5]. Yifan Chen & Vimal K. Dubey. “Accuracy of Geometric Channel–
Modeling Methods”. IEEE Transactions on Vehicular Technology.
Vol. 53, No. 1, Pages: 82 – 93, January 2004.
[6]. Jon Wallace, Huseyin Ozcelik, Markus Herdin, Ernst Bonek &
Michael Jensen. “Power and Complex Envelope Correlation for
Modeling Measured Indoor MIMO Channels: A Beamforming
Evaluation”. The IEEE 58th Vehicular Technology Conference
2003. VTC 2003 –Fall. Vol. 1, Pages: 363 – 367, 6 – 9 October
2003. Orlando, Florida, U. S. A.
[7]. Ramón Parra–Michel, Valeri Ya. Kontorovitch & Aldo G. Orozco–
Lugo. “Modeling Wideband Channels using Orthogonalizations”.
IEICE Transactions on Electronics. Vol. E85 – C, No. 3, Pages:
544 – 551. March 2002.
[8]. Kun–Wah Yip & Tung–Sang Ng. “Karhunen–Loève Expansion of
the WSSUS Channel Output and Its Application to Efficient
Simulation”. IEEE Journal on Selected Areas in Communications.
Vol. 15, No. 4, Pages: 640 – 646, May 1997.
[9]. Alberto Alcocer–Ochoa, Ramón Parra–Michel & Valeri Ya.
Kontorovitch. “Wideband–MIMO Channel Simulation using
Orthogonalizations”. World Wireless Congress 2004, WWC 2004,
May 25 – 28, 2004. San Francisco, California, U. S. A.
[10]. Akbar M. Sayeed. “Decosntructing Multiantenna Fading
Channels”. IEEE Transactions on Signal Processing. Vol. 50, No.
10, Pages: 2563 – 2579, October 2002.
[11]. Werner Weichselberger, Huseyin Ozcelik, Markus Herdin & Ernst
Bonek. “A Novel Stochastic MIMO Channel Model and Its
Physical Interpretation”. International Symposium on Wireless
Personal Multimedia Communications, WPMC 2003. Yokosuka,
Japan, October 2003.
[12]. Ping Zhao, Guihong Liu & Chun Zhao. “A Matrix–Valued KL–like
Expansion for Wide–Sense Stationary Random Processes”. IEEE
Transactions on Signal Processing. Vol. 52, No. 4, Pages: 914 –
920, April 2004.
[13]. A. Juravlev, V. Khlebnikov & V. Kontorovitch. “Adaptive
Radiosystems with Antenna Arrays”. Leningrad University Press,
1991. (In Russian.)
[14]. Harry L. Van Trees. “Detection, Estimation and Modulation
Theory”. Part IV: Optimum Array Processing. John Wiley & Sons.
2002.
[15]. Richard B. Ertel, Paul Cardieri, Kevin W. Soberby, Theodore S.
Rappaport & Jeffrey H. Reed. “Overview of Spatial Channel
Models for Antenna Array Communications Systems”. IEEE
Personal Communications. Pages: 10 – 22, February 1998.
[16]. Richard B. Ertel & Jeffrey H. Reed. “Angle and Time of Arrival
Statistics for Circular and Elliptical Scattering Models”. IEEE
Journal on Selected Areas in Communications. Vol. 17, No. 11,
Pages: 1829 – 1840, November 1999.
[17]. S. S. Mahmoud, Z. M. Hussein & Peter O’Shea. “A Space – Time
Model for Mobile Radio Channel with Hiperbolically Distributed
Scatters”. IEEE Antennas and Wireless Propagation Letters. Vol. 1,
Pages: 211 – 214, 2002.
[18]. Alberto Alcocer–Ochoa, Ramón Parra–Michel & Valeri Ya.
Kontorovitch. “Wideband MIMO Channel Model Based on
Geometrical Approximations”. I International Conference on
Electric and Electronic Engineering and X Conference on Electrical
Engineering 2004, ICEEE–CIE 2004. September 8 – 10, 2004.
Acapulco, Guerrero, Mexico.
[19]. Kai Yu & Bjorn Ottersten. “Models for MIMO Propagation
Channels, a Review”. 2002–07–08 in Wiley Journal on Wireless
Communications and Mobile Computing Special Issue on Adaptive
Antennas and MIMO Systems. IR–S3–SB–0223.
[20]. J. A. Straton, P. M. Morse, L. J. Chu, J. D. C. Little & F. J. Corbató.
“Spheroidal Wave Functions”. The Technology Press of MIT and
John Wiley & Sons, New York, 1956.
[21]. Guangze Zhao & Sergei Loyka. “Impact of Multipath Clustering on
the Performance of MIMO Systems”. The IEEE Wireless
Communications and Networking Conference, WCNC 2004. Vol. 2,
Pages: 765 – 770, March 21 – 25, 2004.
[22]. Ramón Parra–Michel, Valeri Ya. Kontorovitch & Aldo G. Orozco–
Lugo. “Simulation of Wideband Channels with Non–Separable
Scattering Functions”. The IEEE Proceedings of the International
Conference on Acoustic, Speech and Signal Processing, ICASSP
2002. Pages III.2829 – III.2832. May 13 – 17, 2002. Orlando,
Florida, U. S. A.
[23]. F. R. Gantmacher. “Matrix Theory”. American Mathematical
Society, 1998.
[24]. Robert J. Piechocki, Joe P. McGeehan & George V. Tsoulos. “A
New Stochastic Spatio–Temporal Propagation Model (SSTPM) for
Mobile Communications with Antenna Arrays”. IEEE Transactions
on Communications. Vol. 49, No. 5, Pages: 855 – 862, May 2001.
[25]. 3GPP and 3GPP2, “SCM–103: Spatial Channel Model AHG”,
January 2003.
[26]. Peter J. Smith & Mansoor Shafi. “The Impact of Complexity in
MIMO Channel Models”. IEEE International Conference on
Communications, 2004. Vol. 5, Pages: 2924 – 2928, 20 – 24 June
2004.
[27]. Serguei Primak, Valeri Ya. Kontorovitch & Vladimir Lyandres.
“Stochastic Methods and Their Applications to Communications:
Stochastic Differential Equations Approach”. John Wiley & Sons,
2004.
[28]. Harry L. Van Trees. “Detection, Estimation and Modulation
Theory”. Part I: Detection, Estimation and Linear Modulation
Theory. John Wiley & Sons. 1968.
[29]. W. C. Jakes. “Microwave Mobile Communications”. John Wiley &
Sons, New York, 1979.
[30]. J. Salz & J. H. Winters. “Effect of Fading Correlation on Adaptive
Arrays in Digital Mobile Radio”. IEEE Transactions on Vehicular
Technology. Vol. 42, No. 4, Pages: 1049 – 1057. November 1994.